Given \( 2+9+16+\ldots \) (to \( 251^{\text {si }} \) terms) Determine the sum of the terms. Write in sigma notation How many terms are divisible by 4 .

Let's analyze the given arithmetic series and address each part of the problem step by step. ### 1. Understanding the Series The series provided is: \[ 2, \, 9, \, 16, \, \ldots \] Let's identify the pattern: - **First Term (\(a\))**: \(2\) - **Common Difference (\(d\))**: \[ 9 - 2 = 7 \] \[ 16 - 9 = 7 \] Thus, it's an arithmetic sequence with: \[ a = 2 \] \[ d = 7 \] \[ n = 251 \text{ terms} \] ### 2. Writing the Sum in Sigma Notation The general form of the \(k\)-th term in an arithmetic sequence is: \[ a_k = a + (k-1)d \] Substituting the known values: \[ a_k = 2 + (k-1) \times 7 = 7k - 5 \] Therefore, the sum of the first 251 terms can be written in sigma notation as: \[ \sum_{k=1}^{251} (7k - 5) \] ### 3. Calculating the Sum of the Series While the problem primarily asks for the sigma notation and the count of terms divisible by 4, it's also useful to know how to compute the sum. The sum of an arithmetic series is given by: \[ S_n = \frac{n}{2} \times (a + l) \] where: - \( l \) is the last term. First, find the last term (\( l \)): \[ l = a + (n-1)d = 2 + (251-1) \times 7 = 2 + 250 \times 7 = 2 + 1750 = 1752 \] Now, compute the sum: \[ S_{251} = \frac{251}{2} \times (2 + 1752) = \frac{251}{2} \times 1754 = 251 \times 877 = 220,327 \] ### 4. Determining How Many Terms Are Divisible by 4 We need to find the number of terms in the series that are divisible by 4. Each term in the series is given by: \[ a_k = 7k - 5 \] We require: \[ 7k - 5 \equiv 0 \pmod{4} \] Simplifying: \[ 7k \equiv 5 \pmod{4} \] Since \(7 \equiv 3 \pmod{4}\) and \(5 \equiv 1 \pmod{4}\), the equation becomes: \[ 3k \equiv 1 \pmod{4} \] Multiplying both sides by the inverse of 3 modulo 4 (which is 3): \[ k \equiv 3 \times 1 \equiv 3 \pmod{4} \] Thus, \(k\) must be of the form: \[ k = 4m + 3 \quad \text{for some integer } m \] To find the number of such \(k\) within 1 to 251: \[ 4m + 3 \leq 251 \implies 4m \leq 248 \implies m \leq 62 \] Since \(m\) starts from 0: \[ m = 0, 1, 2, \ldots, 62 \] Total number of terms divisible by 4: \[ 63 \text{ terms} \] ### **Final Answer** - **Sigma Notation**: \[ \sum_{k=1}^{251} (7k - 5) \] - **Number of Terms Divisible by 4**: \[ 63 \text{ terms} \]